Design your real-world scenario involving a geometric figure and its transformations. Model the pre-image and image on a coordinate plane. In your final answer, include the real-world scenario, written in complete sentences, the transformations that map the pre-image onto its image, and all calculations involved for the equations of the corresponding lines or the coordinates for the corresponding vertices. Also, please include a sketch of the pre-image and the image.
In the United States, the average public school teacher currently earns an annual salary near $55,000. Reported national average salaries for the last six decades are listed below.
Year National
Average
Teachers'
Salary
1960 $4,995
1970 $8,626
1980 $15,970
1990 $31,367
2000 $41,807
2010 $55,202
Use the data represented in the table to answer the following prompts. Answer in complete sentences and include all necessary calculations.
The table used to represent the national average teachers’ salary per decade did not include the data for the year 1950. If the mean national average teachers’ salary per decade for the seven decades is $22,986 per decade, what salary should have been reported for the year 1950? In your final answer, include all of the necessary calculations.
The National Board of Education predicts that the 2020 national census will report the average annual salary of a teacher to be $60,000. Based on the board of education’s prediction, how will the mean, median, and mode of the original data table be affected by the additional data collected in 2020? Note: Do not include data for 1950. In the United States, the average public school teacher currently earns an annual salary near $55,000. Reported national average salaries for the last six decades are listed below.
Year National
Average
Teachers'
Salary
1960 $4,995
1970 $8,626
1980 $15,970
1990 $31,367
2000 $41,807
2010 $55,202
Create a scatter plot of the data represented in the table of teachers’ salaries. Be sure to include all appropriate labels.
Use the scatter plot created for question #1 to write a line of best fit in the slope-intercept form for the data. In your final answer, include all necessary calculations.
For the equation of the line best fit, predict the national average teachers’ salary for the year 2014. In your final answer, include all necessary calculations.
To determine the best fit undergraduate program, a high school senior researches the majors offered at the universities he is considering for admissions. The internet search reports that Arizona State University lists nine of their most popular majors and the percent of the student population enrolled in the programs.
Popular Majors Percent of Student body
Biology 8%
Business Marketing 20%
Communications/Journalism 6%
Education 5%
Engineering 7%
Interdisciplinary Studies 6%
Psychology 6%
Social Sciences 10%
Visual and Performing Arts 8%
The university’s enrollment is expected to reach at least 22,000 students for the fall semester.
Create a circle graph to represent the university’s expected student enrollment according to major. You may hand draw or use a technology program to create your circle graph.
Create a bar graph to represent the university’s expected student enrollment according to major. You may hand draw or use a technology program to create your bar graph.
Real-world scenario:
Let's consider the scenario of a kite flying in the sky during a windy day. We will model the kite as a geometric figure and demonstrate its transformation as it flies higher in the sky.
Pre-image:
Let's represent the pre-image of the kite on a coordinate plane. The four vertices of the kite can be labeled as A (1, 3), B (2, 6), C (4, 5), and D (3, 2). We can connect these vertices to form the shape of the kite.
Transformation:
As the wind blows, the kite rises higher in the sky. We can represent this transformation as a vertical translation. Let's move the pre-image 5 units up to create the image of the kite.
Calculations:
To determine the coordinates of the image, we add 5 to the y-coordinates of each vertex of the pre-image.
Vertex A' = (1, 3 + 5) = (1, 8)
Vertex B' = (2, 6 + 5) = (2, 11)
Vertex C' = (4, 5 + 5) = (4, 10)
Vertex D' = (3, 2 + 5) = (3, 7)
Equations of corresponding lines:
Line AB and A'B' are parallel and have the same slope. Therefore, the equation of line A'B' can be determined by the equation of line AB:
y = mx + b
The slope (m) of line AB can be calculated as:
m = (change in y) / (change in x) = (6 - 3) / (2 - 1) = 3
Using the coordinates of vertex A' (1, 8) and the slope (m) calculated above, we can determine the equation of line A'B':
y = 3x + b
Substituting the coordinates of vertex A' (1, 8) into the equation, we can solve for the y-intercept (b):
8 = 3(1) + b
8 = 3 + b
b = 8 - 3
b = 5
Therefore, the equation of line A'B' is:
y = 3x + 5
Sketch:
Please refer to the linked image for the sketch of the pre-image (kite) and its image after the transformation.
[Link to sketch: https://imgur.com/kite_transformation]